1、_ COPVNASASP-3004iii_,NATIONAL AERONAUTICS AND SPACE ADMINISTRATIONProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-FriProvided by IHS Not for ResaleNo reproduction or networking permitted without license from IHS-,-,-ForewordTHIs REPORT PRESENTS, in
2、tabular form, the results ofthe calculation of supersonic flow fields about right circular cones at zero angleof attack. These calculations were performed using the Taylor and Maceolltheory. Numerical integrations were performed using the Runge-Kutta methodfor second-order differential equations.Res
3、ults were obtained for cone angles from 2.5 to 30 in regular incrementsof 2.5 . For each of these 12 cone angles, a series of 16 problems was computedat nominal free-stream Mach numbers from 1.5 to 20.0. The free-stream Machnumber was not increased in even increments, but the same values were usedfo
4、r each cone angle. In all calculations, the desired free-stream Mach numberwas obtained to six or more significant figures.The data listed in this report ard essentially the same as those of Zden_kKopals Tables of Supersonic Flow Around Cones (M.I.T. Tech. Rep. No. 1,1947). They differ from Kopals w
5、ork only in the manner of presentation andby the use of a specific heats ratio of 1.4 instead of 1.405. This report repre-sents a complement to NASA SP-3007 in which the flow field about cones atsmall angles of attack in a body-fixed coordinate system is tabulated.lo111Provided by IHSNot for ResaleN
6、o reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ContentsPAGEFOREWORD . IIIINTRODUCTION 1SYMBOLS . 2SOLUTION OF THE EQUATIONS . 3DISCUSSION OF TABLES . 4REFERENCES 5TABLEI,-VALUES FOR
7、 MINIMUM FREE-STREAM MACH NUMBER 62.-VALUES OF P,/P 73.-VALUES OF Ps/9 . 84.-VALUES OF TJT 95.-VALUES OF AS/R 106.-VALVES OF M* AT CONE SURFACE 117.-VALUES OF M AT CONE SURFACE 128.-VALUES OF fl AT CONE SURFACE . 139.-VALVES OF SURFACE PRESSURE COEFFICIENT . 1410.-SHOCK=WAVE RESULTS FOR MINIMUM FREE
8、-STREAM MACHNUMBER .11.-VALUES OF SHOCK WAVE ANGLE 8_12.-VALUES13.-VALUES14 .-VALUES15.-VALUES16._VALUES17.-VALUES151617OF P_/P OF pw/p . 18OF T./T . 19OF M_, IMMEDIATELY BEHIND SHOCK WAVE . 20OF M IMMEDIATELY BEHIND SHOCK WAVE 21OF #, IMMEDIATELY BEHIND SHOCK WAVE 22VProvided by IHSNot for ResaleNo
9、 reproduction or networking permitted without license from IHS-,-,-TAB LE S18- 34.-0,= 2.5;35- 51.-0,= 5.0o;52- 68.-0,= 7.5o;69- 85.-0,= lO.O;86-102.-0,= 12.5;103-119.-8,=15.0;120-136.-#,= 17.5 ;137-153.-#,=20.0;154-170.-#_=22.5 ;171-187.u#,=25.0;188-203.-8,=27.5;204-219.-8,=30.0;ConalM-1.0121844M=1
10、.0383341M=1.0735583M=1.1159051M=1.1643198M-1.2182190211/.=1.2773745M=1.3419094M=1.4123337M=1.4895952M=1.5751393M=1.6710795Flow FieldPAGESto 20.0 23-56to 20.0 57-90to 20.0 91-124to 20.0 125-158to 20.0 159-192to 20.0 193-225to 20.0 226-258to 20.0 259-291to 20.0 292-324to 20.0 325-357to 20.0 358-389to
11、20.0 390-421viFi!iilI!l !Provided by IHS Not for ResaleNo reproduction or networking permitted without license from IHS-,-,-IntroductionTHE SOLUTION of supersonic flowfields by the method of characteristics requiresthat the flow conditions along a starting line inthe flow field be known. For sharp-n
12、osed bod-ies of revolution, this information is usuallyobtained from the solution of the flow fieldabout circular cones. During the process ofsetting up a program for treating bodies ofrevolution by the method of characteristics, itwas decided to compute the starting flow fieldrather than use the ta
13、bles published by Kopalin references 1 and 2.With these programs available, it appeareddesirable to prepare a set of cone tables forcones at small angles of attack in a body-fixedcoordinate system. In order not to restrict theMach numbers to those of references 1 and 2,this required also a set of co
14、ne tables for thecase of zero angle of attack. This latter set ispresented in this report; the angle-of-attackcase will be covered in reference 3. Thispresent set of tables differs from those ofreference 1 only in the manner of presentationand the value of specific heats ratio “y. In allof the prese
15、nt calculations, the ideal gas valueof _= 1.4 has been usedlOne of the uses envisioned for the results con-tained herein is in types of solutions of whichthe shock-expansion theories are typical. Thus,the minimum cone angle was 2.5 and this wasincreased by increments of 2.5 to a maximumcone angle of
16、 30 (a total of 12 cone angles).For each of the cone angles, results were com-puted for a constant series of free-stream Machnumbers from 1.5 to 20. In addition, a solu-tion was computed which yielded the minimumfree-stream Mach number for a completelysupersonic conical flow field (u,=_/_)- Thiswas
17、the lowest value of M for which anysolutions were obtaine x-axis=cone axisR universal gas constantAS increase in entropyT absolute temperatureu, v velocity components, dimensional (fig. 1)u nondimensional velocity along conical rayline in spherical coordinate system(fig. 1), u/V,v nondimensional vel
18、ocity normal to con-ical ray line in spherical coordinatesystem, _/_7_V resultant nondimensional velocity atany conical ray line, _/_zV velocity (dimensional)_t limiting velocity due to expansion intoa vacuumratio of specific heats, cp/co; ideal gasvalue = 1.4Mach angle_b, flow direction angle, angl
19、e between velo-city vector V and cone axisv density0 conical ray angle, from cone axisSubscripts:s denotes values at cone surfacecoconditions back of shock wavefree-stream conditionM_ShockShockWaVeFIGVR_ 1.-Coordinate System.line-Characteristiclinerlii1Provided by IHSNot for ResaleNo reproduction or
20、 networking permitted without license from IHS-,-,-Solution of theTHE DERIVATIONS of the basicequations for the conical flow problem aregiven quite adequately in reference 1 and willnot be repeated here. The differential equa-tion that is the formulation of the conicalflow problem in a spherical coo
21、rdinate system(fig. 1) is:d_u . a_(u+v cot 0)dO2 -f-u= v-_-_ . (1)whereandduv=_-_ (2)a2-_-_ (1-u2-V _) (3)In the foregoing equations, all velocities arenondimensionalized, as in reference 1, bydividing them by the limiting velocity attain-able by adiabatic expansion into a vacuum.This system of comp
22、utation was used, eventhough the results are later transposed intoanother reference system, in order to make useof the parameters in reference 1 as convenientguides in setting up the numerical calculations.Boundary conditions must be prescribedalong with equations (1) to (3), and they are(4)at the s
23、urface of the cone. The upper boundarycondition is found by requiring the results ob-tained from the integration of equation _1) tosatisfy the Rankine-Hugoniot equations whichcan be expressed astan O_ “Y-1 u2-1 (5)_+ 1 uvWhen equation (5) is Fulfilled by the resultsfrom equation (1) the free-stream
24、.Mach numberis given byEquationsM / 2 u 2_“VT-1 cos 2 O-u s (6)The solutions of equation (1) presentedherein were obtained using the Runge-Kuttaintegration method. Computation was startedat the solid surface of the cone by specifying avalue of u, and ended when the shock-waveconditions were satisfie
25、d. Integration stepsize investigations, which are not includedherein, indicate that the maximum integrationerror is less that 5X10 -8 at any point in anysolution. The Rankine-Hugoniot conditionswere satisfied to 2 X 10 -7.The method of solution outlined does notyield a desired arbitrary frec-stream
26、Machnumber without a priori knowledge of thevalue of u, that should be specified. Therefore,an iteration on u_ was included in the procedurein order that both cone angle and free-streamMach number could be specified.Integration of equation (1) yields values ofu, v, and a at each conical ray angle 0.
27、 Theseresults are transformed into the more usableforms of M*, _1, and _ in the following manner:M, V /_,+1, =h_= _/_:-_ (u_+e) (7)sinceand the flow direction angle 1 isFurthermore,l=0 + tan_ 1v_t_= sin-_/ a_u_+v _However, due to space limitations, the valuesof the Mach angle g will be presented _th
28、 thesmall-angle-of-attack results.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4DiscussionIT HAS BEEN ASSUMED that allof the cones are terminated at a base diameterof unity and the geometric location of theleft-running characteristic line emanatin
29、g fromthis point has been computed. These resultsare given in an x,r cylindrical coordinatesystem that has its origin at the cone vertex.Furthermore, this characteristic line was builtup using the basic inte.gration step size duringthe integration and cannot be duplicatedexactly using the tabulated
30、values. By meansof a base diameter ratio it is possible to pro-portion the locations of these characteristicpoints on a cone of any other size. Of course,._y/_, 1, and u stay constant since the conicalray angles do not change.The tables for the individual solutions wereprinted directly from the mach
31、ine calculationsand were not converted to decimal form. Theleading sign is the algebraic sign of the quantity.The next eight digits that represent thesize of the quantity are considered to be0.xxxxxxxx. Coming last are an algebraicsign and two digits that are the exponent of10 _-xXby which the size
32、of the quantity mustbe multiplied to obtain the correct decimalpoint location. Thus a number that is printedin the machine code as -47168732+00 isread as -0.47168732. The angular incrementsat which the tables are printed generally aredecreased as the flow field is traversed. Thiswas done to keep the
33、 incremental distancealong the characteristic line from increasingcontinuously. In the tables, the quantitiesM*, , x, and r are listed as functions of theconical ray angle 8, starting at the cone surfaceand terminating at the shock wave; x and rare the coordinates (in the cylindrical coordinateof Ta
34、blessystem) of the characteristic starting line incone base calibers while M* is the resultantnondimensionalized velocity and 1 is the flowdirection angle in radians. The end resultsgiven with each table (AS/R, P,/P, p,/p,and T,/T._) were computed using the integratedresults with shock wave and isen
35、tropic equations(ref. 4). The summary tables for surfaceconditions were tabulated by hand from themachine printed individual tables and aregiven in decimal form.Table 1 gives a summary of the surfaceresults of the calculations for the minimumfree-stream Mach number. In tables 2 through9 are summariz
36、ed some of the surface resultsfor the cones in which the free-stream Machnumber was specified. These results are tabu-lated for each cone angle as a function of thenominal free-stream Mach number. The itera-tion procedure mentioned earlier producedfree-stream Mach numbers for all solutionsthat corre
37、spond to the nominal values to six ormore significant figures. Shock-wave anglesand flow results immediately behind the shockwave are given in table 10 for the minimumfree-stream Mach number solutions. Tables11 through 17 give shock-wave angles and flowparameters immediately behind the shockwave for
38、 all solutions in which the free-streamMach number was specified. Results for thecomplete flow field of each calculation aregiven in tables 18 through 219. In thesetables, the results are tabulated starting at thecone surface and proceeding through the flowfield to the shock wave. The free-streamMac
39、h number listed with each of these tablesis the exact value computed from the Rankine-Hugoniot equations.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-References1. KOPAL, ZDEN_K: Tables of Supersonic Flow Around Cones. Massachusetts Instituteof Tec
40、hnology, Tech. Rep. No. 1, 1947.2. KOPAL, ZDE.X_K: Tables of Supersonic Flow Around Yawing Cones. MassachusettsInstitute of Technology, Tech. Rep. No. 3, 1947.3. SIMs, JosEP_ L.: Tables for Supersonic Flow Around Right Circular Cones at SmallAngle of Attack. NASA SP-3007, 1964.4. AMES RESEARCH STAFF
41、: Equations, Tables and Charts for Compressible Flow. NACARep. 1135, 1953.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-oE 008Q.Q.8m_0 “_ o _ _ _ _ _ _ _ _._ _00 0 Q _ _. _ _ _ _ _ o _ _, , .,4 ,J ,4 ,4 ,J A ,4 _ _ ,J ,-iU_ 0 U_ 0 U% 0 _e_ 0 _ 0 U_
42、 0_ _,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-80000U_0Lt_0U_r_w_b-“_ _ 0 ,r_ 0 _r_ o _ o o o o o o oProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-eE-000kOfl _ _ _ _ _ _ _ _ o _ _o. o _ o _ _
43、-t. _ g_ t“-t“-0_ _ o _o _ _ D _ _ _ _ oo o o ooooo ooProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8_L-.-4,rc_mo ot_c_J00 oo%o%oooooc_ _ o _ o u_ o _ o o o o o o oo.Provided by IHSNot for ResaleNo reproduction or networking permitted without licen
44、se from IHS-,-,-188IVl40 ooo !rl ,-1 _-I i-I rl ,_1 I-I rl ,-I ,-I ,-I ,“1 0,1 0,1 _,0 o_ _ _ _ ,-.t _ _ ,.-i _ _ _ _ _ o_ 0_ _ _ _ _ _ _ _ _ _ _ ,-i _ _,ijProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- 19,4.%i, ,_,|o0oOoc_,-IL_%t_- “Ooo _i_ _ o_r/
45、 tr_ _ o _ o ux o ux o o o o o o o oProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-20.,-q00mt_CoIfX_, _ o , o , o. , o ,_ _o o o o. oProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2100r,jZ)oo%oo%00%
46、,%Oo.%oo,_ ,-i,-i,-i._ _ _ cu c_ _ _ _ _ _ ,-i_4 cu c_ _ _ _ _ _ _ ._ _ _, ,-i_ ,_ _ c,i_ _ _-,_ _ _ _ _ _ _ _,-i,-i _ _ _ _ _ _ _ _i_ _ o ,c,o _ o _ o o o o o o o oProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-220.,.-4000t_0,=,=!ii!i!l!i!ii!I!I!i!
47、iiI!il_(_o_o_o_ooo_,_Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TABLE 18.-Conical Flow Field. 0,=2.5; .7tI,_=1.0121844 23+40825000+00 +10121844+01 +43633275-01 +14174491018 M_ i x r+43633275-01+61086567-01+78539851-0“1+95993134-01+11344642+00+13089970+00+14835298+00+16580626+00+18325955+00+20071283+00+21816611+00+23561940+00+25:307268+00+27052596+00+28797925+00+50543253+00+32288
